### Maass forms at high eigenvalue

The image used at the top of each page in this
website is a density plot of an eigenfunction
of the Laplacian on the upper half-plane with the hyperbolic
(constant negative-curvature) metric,
obeying certain symmetries which make it a so-called *arithmetic surface*.
Black shows large values of the square of the mode value, white
zero.
Its fundamental domain is the
strip shown, bounded by an arc at one end and unbounded on the other.
For aesthetic reasons the *y*-axis has been chosen horizontal;
in Cartesian coordinates the area element is *dxdy/y*^{2}.
These eigenfunctions are known as *Maass forms*. They have importance
in number theory, as well as providing a rare example where QUE has
been (recently) proven.

An introduction can be found in the Baltimore review article of
Peter Sarnak, here.

Below is a set of 12 consecutive such eigenfunctions, with mode number *n*=5601, 5602,..., 5612, shown in the more usual
orientation (*y* is vertical, and the *x*-axis is visible as the
line segment at the bottom), and with the density now with the
correct metric (intensity proportional to square of mode value times
1/*y*^{2}).
The modes were evaluated by Alex Barnett using code, and
Fourier coefficients, provided by Holger Then.
Click on image to enlarge: